Input–Output Theory

Input–output theory relates a localized quantum system to traveling fields incident on it and emitted from it. It is the language behind cavity reflection, transmission, homodyne measurement, photon counting, circuit-QED readout, and many Markovian quantum-optics experiments. When the ports and drives are chosen to stabilize useful states, the same formalism becomes a workhorse for Reservoir Engineering.

Here we connect the general field boundary relation to arbitrary system coupling operators and continuous measurement. For cavity linewidths, critical coupling, ringdown, and fitting reflection or transmission data, see Input–Output Theory Overview.

The simplest slogan is:

output field = input field + field radiated by the system

For a single Markovian port with coupling operator $L$, a common convention is

$$b_{\mathrm{out}}(t) = b_{\mathrm{in}}(t) + L(t).$$

For a cavity mode coupled to one port with $L=\sqrt{\kappa}\,a$,

$$b_{\mathrm{out}}(t) = b_{\mathrm{in}}(t) + \sqrt{\kappa}\,a(t).$$

Some references use the opposite sign for the radiated term. The sign depends on phase conventions for the port field and coupling Hamiltonian; measurable spectra and fluxes are unchanged when the convention is used consistently.

Field Normalization

The input field $b_{\mathrm{in}}(t)$ is a traveling-wave annihilation operator normalized so that

$$[b_{\mathrm{in}}(t),b_{\mathrm{in}}^\dagger(t')] = \delta(t-t').$$

Thus $b_{\mathrm{in}}$ has units $\mathrm{s}^{-1/2}$, unlike a dimensionless intracavity annihilation operator $a$. For a field with finite normally ordered intensity, the photon flux has units of inverse time:

$$\Phi_{\mathrm{in}}(t) = \langle b_{\mathrm{in}}^\dagger(t)b_{\mathrm{in}}(t)\rangle.$$

For vacuum input,

$$\langle b_{\mathrm{in}}(t)b_{\mathrm{in}}^\dagger(t')\rangle = \delta(t-t'), \qquad \langle b_{\mathrm{in}}^\dagger(t)b_{\mathrm{in}}(t')\rangle = 0.$$

For a thermal input with occupation $\bar n$,

$$\langle b_{\mathrm{in}}^\dagger(t)b_{\mathrm{in}}(t')\rangle = \bar n\,\delta(t-t'),$$

and

$$\langle b_{\mathrm{in}}(t)b_{\mathrm{in}}^\dagger(t')\rangle = (\bar n+1)\delta(t-t').$$

These delta correlations describe a white-field approximation over the system bandwidth. Do not set $t=t'$ in the thermal correlation and interpret $\bar n\delta(0)$ as a finite photon flux. A physical detector has a frequency band $\mathcal B$ and, for unit transmission there, measures thermal flux $\int_{\mathcal B}\bar n(\omega)\,d\omega/(2\pi)$. Mode occupation and photon flux are different quantities.

Langevin Equation

For one cavity mode with Hamiltonian $H=\hbar\omega_c a^\dagger a$ and coupling $L=\sqrt{\kappa}\,a$, the Heisenberg–Langevin equation in the same convention is

$$\dot a = i[H/\hbar,a] - \frac{\kappa}{2}a - \sqrt{\kappa}\,b_{\mathrm{in}}(t).$$

Since

$$i[H/\hbar,a] = -i\omega_c a,$$

one obtains

$$\dot a = - \left( i\omega_c+\frac{\kappa}{2} \right)a - \sqrt{\kappa}\,b_{\mathrm{in}}(t).$$

The noise term is not optional. Without the input field, damping would not preserve the correct commutation relations.

Master Equation Connection

For vacuum incident on a Markov port, ignoring the output gives

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \mathcal D[L]\rho,$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12\{L^\dagger L,\rho\}.$$

For a cavity port with $L=\sqrt{\kappa}a$,

$$\mathcal D[L]\rho = \kappa\mathcal D[a]\rho.$$

A thermal input with approximately constant occupation across the relevant band instead gives $(\bar n+1)\mathcal D[L]\rho+\bar n\mathcal D[L^\dagger]\rho$. A coherent input adds the drive Hamiltonian below after displacing its mean. The state of the input is therefore part of the model, not something fixed by the boundary relation alone.

For a vacuum port, the same coupling operator appears in three related descriptions:

unconditional master equation:  D[L] rho
input–output boundary:          b_out = b_in + L
trajectory measurement:         output record determined by L

This is why collapse operators must be interpreted with their physical ports and measurement schemes specified.

Multiple Ports

If several independent ports couple to the same system, each has its own input and output field:

$$b_{j,\mathrm{out}}(t) = b_{j,\mathrm{in}}(t) + L_j(t).$$

For independent vacuum inputs, the unconditional master equation contains

$$\sum_j \mathcal D[L_j]\rho.$$

For a two-sided cavity,

$$L_1=\sqrt{\kappa_1}\,a, \qquad L_2=\sqrt{\kappa_2}\,a,$$

and internal loss can be represented by another unmonitored channel

$$L_{\mathrm{int}} = \sqrt{\kappa_{\mathrm{int}}}\,a.$$

The total energy-decay rate is

$$\kappa = \kappa_1+\kappa_2+\kappa_{\mathrm{int}},$$

but only monitored ports contribute to measured output records.

Coherent Drives

A coherent incident field can be written as a mean amplitude plus quantum noise:

$$b_{\mathrm{in}}(t) = \beta_{\mathrm{in}}(t) + \xi_{\mathrm{in}}(t),$$

where $\xi_{\mathrm{in}}$ has vacuum correlations for a coherent state. A displaced thermal state instead has thermal correlations. For a cavity, the mean drive enters the equation of motion as

$$\dot a = - \left( i\omega_c+\frac{\kappa}{2} \right)a - \sqrt{\kappa}\,\beta_{\mathrm{in}}(t) - \sqrt{\kappa}\,\xi_{\mathrm{in}}(t).$$

Equivalently, the coherent drive may be moved into an effective Hamiltonian,

$$H_{\mathrm{drive}} = i\hbar \left( \sqrt{\kappa}\,\beta_{\mathrm{in}}^*a - \sqrt{\kappa}\,\beta_{\mathrm{in}}a^\dagger \right),$$

with the remaining input treated as noise. Different phase conventions move minus signs between $\beta_{\mathrm{in}}$, $H_{\mathrm{drive}}$, and the output relation.

Reflection as an Interference Check

For a lossless cavity with a single port, detuning $\Delta=\omega-\omega_c$, and energy linewidth $\kappa$, the steady reflected amplitude is

$$r(\omega)=1-\frac{\kappa}{\kappa/2-i\Delta}.$$

It has unit magnitude and gives $r=-1$ on resonance: all incoming energy returns, with a phase shift. The result follows from interference between the incident and cavity-emitted fields. Treating their intensities as independent positive contributions would violate flux conservation. The cavity calculation extends this example to loss and multiple ports.

Output Flux

The output photon flux is

$$\Phi_{\mathrm{out}}(t) = \langle b_{\mathrm{out}}^\dagger(t)b_{\mathrm{out}}(t)\rangle.$$

Using $b_{\mathrm{out}}=b_{\mathrm{in}}+L$ gives

$$\Phi_{\mathrm{out}} = \langle b_{\mathrm{in}}^\dagger b_{\mathrm{in}}\rangle + \langle L^\dagger L\rangle + \langle b_{\mathrm{in}}^\dagger L\rangle + \langle L^\dagger b_{\mathrm{in}}\rangle.$$

For vacuum input with no coherent amplitude, the interference terms vanish in the usual Markov treatment, and the emitted flux is

$$\Phi_{\mathrm{emit}} = \langle L^\dagger L\rangle.$$

For a driven cavity, the interference terms are essential. Reflection and transmission are field-amplitude phenomena, not just photon-number addition.

Homodyne and Heterodyne Records

Input–output theory supplies the field that is measured by homodyne or heterodyne detection. For a monitored port with vacuum input and coupling $L$, a homodyne quadrature record has the schematic form

$$dY_t = \sqrt{\eta}\, \langle e^{-i\phi}L+e^{i\phi}L^\dagger\rangle_c\,dt + dW_t,$$

after subtracting known coherent input amplitudes. This is the same structure used in Diffusive Trajectories, with the detector model discussed in Homodyne Detection.

Photon counting instead samples the output intensity. For vacuum input and photon-counting efficiency $\eta$, the expected count increment is

$$\mathbb E[dN_t\mid\rho_c] = \eta\, \langle L^\dagger L\rangle_c\,dt.$$

This is the field-theory origin of Photon Counting and Quantum Jump Trajectories. For two-quadrature diffusive monitoring of the same output field, see Heterodyne Detection.

Cascaded and Feedback Systems

Input–output theory also describes cascaded systems, where the output of one component becomes the input of another. In the simplest Markovian limit, propagation delay is neglected and the connection is one-way. The control-oriented entry point for such routed quantum signals is Coherent Feedback.

If delays, reflections, or finite-bandwidth propagation matter, the dynamics may become non-Markovian. Then the simple boundary relation remains locally meaningful, but the reduced system may require delay equations, explicit waveguide modes, or memory kernels.

Assumptions

The basic Markovian input–output formula assumes:

• a continuum of traveling modes near the system frequency;
• approximately frequency-independent coupling over the relevant bandwidth;
• rotating-wave coupling between system and field;
• short propagation time across the coupling region;
• initially specified input field states;
• no unresolved delayed feedback;
• ports that can be treated as independent channels.

Violating these assumptions does not make input–output language useless, but it does mean the single-line formula may no longer produce a valid Markovian master equation.

Exercises

Output Flux from Vacuum Input

Assume $b_{\mathrm{out}}=b_{\mathrm{in}}+L$ and vacuum input with no coherent amplitude. Show why the emitted flux is $\langle L^\dagger L\rangle$.

Solution

The output flux is

$$\Phi_{\mathrm{out}} = \langle b_{\mathrm{out}}^\dagger b_{\mathrm{out}}\rangle.$$

Substitute the input–output relation:

$$\Phi_{\mathrm{out}} = \langle b_{\mathrm{in}}^\dagger b_{\mathrm{in}}\rangle + \langle L^\dagger L\rangle + \langle b_{\mathrm{in}}^\dagger L\rangle + \langle L^\dagger b_{\mathrm{in}}\rangle.$$

For vacuum input with no coherent amplitude, the normally ordered input flux and the interference terms vanish in the Markov treatment. Therefore

$$\Phi_{\mathrm{emit}} = \langle L^\dagger L\rangle.$$

Account for every output port

A cavity is driven through port 1 and has rates $\kappa_1,\kappa_2,\kappa_{\mathrm{int}}$, with total $\kappa$ their sum. Its coherent amplitudes are

$$r=1-\frac{\kappa_1}{\kappa/2-i\Delta},\qquad t=-\frac{\sqrt{\kappa_1\kappa_2}}{\kappa/2-i\Delta}.$$

Show that the reflected, transmitted, and internally lost power fractions sum to one.

Solution

Let $d=(\kappa/2)^2+\Delta^2$. Then

$$R=1-\frac{\kappa_1\kappa}{d}+\frac{\kappa_1^2}{d},\qquad T=\frac{\kappa_1\kappa_2}{d},\qquad A=\frac{\kappa_1\kappa_{\mathrm{int}}}{d}.$$

Since $\kappa=\kappa_1+\kappa_2+\kappa_{\mathrm{int}}$, $R+T+A=1$. The negative interference contribution in $R$ is essential. Here $A$ counts power deposited in the internal-loss reservoir, whether or not that reservoir is monitored.

Empty-Cavity Reflection

Using

$$r(\omega) = 1 - \frac{\kappa}{\kappa/2-i\Delta},$$

show that a lossless single-sided cavity has $|r(\omega)|=1$.

Solution

Rewrite

$$r(\omega) = \frac{\kappa/2-i\Delta-\kappa} {\kappa/2-i\Delta} = \frac{-\kappa/2-i\Delta} {\kappa/2-i\Delta}.$$

The numerator and denominator have the same magnitude:

$$\left|-\frac{\kappa}{2}-i\Delta\right|^2 = \left|\frac{\kappa}{2}-i\Delta\right|^2 = \frac{\kappa^2}{4}+\Delta^2.$$

Therefore $|r(\omega)|=1$.

Homodyne Signal

For a monitored port with $L=\sqrt{\kappa}a$, write the expected homodyne signal at phase $\phi$ after subtracting known coherent input.

Solution

The record has the form

$$dY_t = \sqrt{\eta}\, \langle e^{-i\phi}L+e^{i\phi}L^\dagger\rangle_c\,dt + dW_t.$$

Substituting $L=\sqrt{\kappa}a$ gives

$$\mathbb E[dY_t\mid\rho_c] = \sqrt{\eta\kappa}\, \langle e^{-i\phi}a+e^{i\phi}a^\dagger\rangle_c\,dt.$$

References

• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010).
• C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Physical Review A 31, 3761–3774 (1985). doi:10.1103/PhysRevA.31.3761.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer (2008).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).